Rueppel's conjecture on the linear complexity of the first $n$ terms of the sequence $(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$ was first proved by Dai using the Euclidean algorithm. We have previously shown that we can attach a homogeneous (annihilator) ideal of $F[x,z]$ to the first $n$ terms of a sequence over a field $F$ and construct a pair of generating forms for it. This approach gives another proof of Rueppel's conjecture. We also prove additional properties of these forms and deduce the outputs of the LFSR synthesis algorithm applied to the first $n$ terms. Further, dehomogenising the leading generators yields the minimal polynomials of Dai.

翻译：Rueppel关于序列$(1,1,0,1,0^3,1,0^7,1,0^{15},\ldots)$前$n$项的线性复杂度猜想最初由Dai利用欧几里得算法证明。我们先前已证明，可为域$F$上序列的前$n$项赋予$F[x,z]$上的齐次（零化子）理想，并为其构造一对生成形式。这一方法给出了Rueppel猜想的新证明。我们还证明了这些形式的其他性质，并推导出应用于前$n$项的LFSR综合算法输出结果。此外，对主生成元进行去齐次化可得到Dai的最小多项式。